Computational Physics

14.2 Derivation of the Navier–Stokes equations 451

by multiplyingEq. (14.1)byvand integrating over the velocity. Using the indices
αandβfor the Cartesian coordinates, we obtain

∂jα

∂t

+

∫

mvα

∑

β

vβ∂βn(r,v,t)d^3 v=

∫

mvα

(

dn

dt

)

collisions

d^3 v, (14.7)

where∂βdenotes a derivative with respect to the coordinaterβ. For the right hand
side, a similar statement can be made as for the equivalent term in the mass equation:
although individual particles involved in a collision change their momenta, thetotal
momentum is conserved on collision. After thus putting the right hand side to zero,
we write (14.7) in shorthand notation as

∂jα

∂t

+∂βαβ=0, (14.8)

where we have introduced the momentum flow tensor

αβ=

∫

mvαvβn(r,v,t)d^3 v, (14.9)

and where we have used the Einstein summation convention in which repeated
indices (in this caseβ) are summed over.
Assuming that we are in equilibrium, we can evaluate the momentum tensor by
substituting forn(r,v,t)the form(14.2):

eqαβ=

∫

vαvβρ(r)exp[−m(v−u)^2 /( 2 kBT)]d^3 v

=ρ(r)(kBTδαβ+uαuβ). (14.10)

This result can be derived by separately consideringα=βandα=β, and working
out the appropriate Gaussian integrals (using the substitutionw=v−u). Noting
thatρkBTequals the pressureP,^1 we arrive at the following two equations:

∂ρ(r,t)

∂t

+∇·j(r,t)=0 (mass conservation); (14.11a)

∂(ρu)

∂t

+∇r·(PI+ρuu)=0 (momentum conservation). (14.11b)

Using the first equation, we can rewrite the second as

∂u(r,t)

∂t

+[u(r,t)·∇r]u(r,t)=−

1

ρ(r,t)

∇rP(r,t). (14.12)

The equations(14.11a)and(14.11b)or(14.12)are theEuler equationsfor a fluid
in equilibrium. These equations neglect dissipative effects.
When the fluid is not everywhere in local equilibrium, the collisions will drive
the system towards equilibrium – hence their effect can no longer be neglected.

(^1) Here, we consider the fluid as an ideal gas; a realistic equation of state may be used instead.